$α, β, γ$ are the roots of the equation $a x^{2} + b x + c = 0$ then frame equation whose roots are $\frac{1}{α}, \frac{1}{β}, \frac{1}{γ}$ .

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Solution

Given, $α, β$ are the roots of the equation $a x^{2} + b x + c = 0$ .
Then form the relation between the roots and co-efficient we get,
$α + β = - \frac{b}{a}$ .....(1) and $α . β = \frac{c}{a}$ .....(2).
Now the equation whose roots are $\frac{1}{α}, \frac{1}{β}$ is
$(x - \frac{1}{α}) (x - \frac{1}{β}) = 0$
or, $x^{2} - x (\frac{α + β}{α . β}) + \frac{1}{α . β} = 0$
or, $x^{2} - x \frac{\frac{- b}{a}}{\frac{c}{a}} + \frac{a}{c} = 0$ [ Using (1) and (2)]
or, $x^{2} + \frac{b x}{c} + \frac{a}{c} = 0$
or, $c x^{2} + b x + a = 0$ .

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